What does the Cartesian product $\mathbb{R} \times \mathbb{R} \times \mathbb{R}$ represent geometrically? Verify that the point $(2, -1, 3)$ belongs to it, and explain why $(0, 0)$ does not.

**Step 1:** $\mathbb{R}\times\mathbb{R}\times\mathbb{R}=\{(x,y,z):x,y,z\in\mathbb{R}\}$ is the set of all ordered triples of real numbers, identified with three-dimensional space (three mutually perpendicular axes through a common origin). **Step 2:** $(2,-1,3)$ has three real entries $2,-1,3$, so it fits the defining pattern $(x,y,z)$ with $x=2,y=-1,z=3$; hence $(2,-1,3)\in\mathbb{R}\times\mathbb{R}\times\mathbb{R}$. **Step 3:** $(0,0)$ has only two entries, so it is an element of $\mathbb{R}\times\mathbb{R}$ (a pair), not of $\mathbb{R}\times\mathbb{R}\times\mathbb{R}$ (which requires exactly three entries) — the two sets consist of objects of a different 'shape' (pairs versus triples) and are not comparable this way. > [!ANSWER] $(2,-1,3)\in\mathbb{R}\times\mathbb{R}\times\mathbb{R}$, since it is an ordered triple of reals; $(0,0)\notin\mathbb{R}\times\mathbb{R}\times\mathbb{R}$, since it has only two components, not three

What does the Cartesian product mathbbR × mathbbR × mathbbR represent geometrically? Verify that the point (2,

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\mathbb{R} \times \mathbb{R} \times \mathbb{R}

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Concept understanding — Cartesian Product of the Reals (R×R and R×R×R)

Taking $A=B=\mathbb{R}$ in the Cartesian-product definition gives $\mathbb{R}\times\mathbb{R}=\{(x,y):x,y\in\mathbb{R}\}$ , written $\mathbb{R}^2$ , which is identified with the familiar coordinate plane — every ordered pair of reals names exactly one point, and every point names exactly one ordered pair. Extending one step further, $\mathbb{R}\times\mathbb{R}\times\mathbb{R}=\{(x,y,z):x,y,z\in\mathbb{R}\}$ , written $\mathbb{R}^3$ , is identified with three-dimensional space in the same way. This is the concrete home in which relations, functions and their graphs (subsets of $\mathbb{R}\times\mathbb{R}$ ) actually live.

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Q.What does the Cartesian product R×R×R\mathbb{R} \times \mathbb{R} \times \mathbb{R}R×R×R represent geometrically? Verify that the point (2,−1,3)(2, -1, 3)(2,−1,3) belongs to it, and explain why (0,0)(0, 0)(0,0) does not.

Concept understanding — Cartesian Product of the Reals (R×R and R×R×R)

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Q.What does the Cartesian product $\mathbb{R} \times \mathbb{R} \times \mathbb{R}$ represent geometrically? Verify that the point $(2, -1, 3)$ belongs to it, and explain why $(0, 0)$ does not.

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